6-Piece Burr Design Definitions

Pieces - each piece must be constructed by removing some of the 12
cubes as numbered in Fig. 1, and the piece must be connected. (Note that
the numbering of the 12 cubes does not match the numbering in my article
in the Journal of Recreational Mathematics . I apologize for this
inconsistency in my notation. This is compounded by the problem of
showing the numbering patterns by `layers.' It may not be clear if the
bottom layer is looked at from the top or bottom.) Fig. 1 is for
length-6 pieces. Length-8 pieces are one unit longer on each end. The
representation of the pieces by which cubes are removed may not be
unique. Two pieces are identical if one can be rotated and/or flipped
end-for-end so as to match the other piece.

Symmetric Piece - a piece is symmetric if an end-for-end flip results
in the same cube configuration. The usual flip would switch cube number
pairs (1,8), (2,7), (3,6), (4,5), (9,12), and (10,11). For `ambiguous'
pieces (see below), other flips are possible.

Assembly - an assembly, or `legal configuration,' is a way of
arranging the six pieces into the 3-dimensional grid of cubes pictured in
Fig. 2 without having two or more pieces occupying the same cube. One
need not be able to physically do this with wooden (rigid) pieces. Two
assemblies are the same if one can be rotated (in three dimensions) so
that it matches the other exactly.

Internal Cubes - cubes in Fig. 2 that may belong to more than one
piece.

External Cubes - cubes in Fig. 2 that belong to only one piece.

Internal Holes - internal cubes not occupied in an assembly. When
discussing the number of `holes' this will be the count referred to.

External Holes - external cubes not occupied in an assembly. For most
analyses in this paper, assemblies with external holes will not be
allowed.

Ambiguous Piece - a piece which can be rotated around its long axis
without creating external holes. In terms of cube numbers from Fig. 1,
such a piece must have either both cubes 1 and 4 present, or both cubes 5
and 8 present. These pieces have been a thorn in the side of 6-piece
burr analyzers and have led to miscommunications because of different
methods of handling.

Solution - a solution is an assembly which can be physically
constructed (or disassembled). The method of assembly or disassembly
must be theoretically correct when used with rigid pieces. (It may not
make use of slightly rounded corners, for example).

`Interesting' Solution - a solution which cannot be split into two
sections on the first move.

Definitions Related to BURR6 Disassembly Method

Rectilinear Solution - a method of assembling or disassembling an
assembly which is carried out by a sequence of moves, each of which is a
linear move of one or more pieces in one of the three directions. The
distances the pieces are moved must be multiples of the unit cube in
length. Solutions of this type are the only ones that the {\erm BURR6}
program can discover. Whether other solutions for holey-6 burrs exist
is, as far as I know, an open question. (See Challenge Questions.)

Displacement - three integers representing the number of units in each
of the three directions that a piece has been shifted from its initial
position. Since constants may have been added to `center' a set of
pieces, these numbers are best viewed in relation to the displacements of
the other pieces in the set.

State - a set of displacements for the pieces in an assembly. Two
states for a set of pieces are the same if the displacements for the
pieces, measured relative to the smallest numbered piece, are the same.
Each new state represents a different geometrical arrangement to which
the pieces of an assembly can be moved.

Subassembly - a set of from two to five pieces that are left together
after an earlier assembly or subassembly has been separated.

Level - during a disassembly process, the number of times that the
direction of movement has changed in order to reach a particular state.
The computer treats all movement in the same direction as being the same
level. The level can be looked on as an estimation of difficulty of the
assembly. It is similar to what would be called `number of moves,' but
is frequently smaller than the count that most people would get when
disassembling a burr. More on this later.